Is intermittent fasting, which does not include water and is intended to reduce lust, considered an imitation of Christian fasting, given that a small amount of food is consumed in the morning as a precaution? And is it necessary to show the family that this small amount of food has been consumed to دفع the شبهة of Christian fasting? And should the sister be informed of this by message, or should one wait until she returns?

There is no harm in refraining from food but not drink with the intention of losing weight, as this does not constitute imitation of Christians. The criterion for blameworthy imitation is imitating disbelievers in matters specific to them. If you were to eat a small amount during the day to avoid the semblance of imitation, it would be good. Fasting the prescribed Islamic fast to reduce the intensity of desire is better, and it is a recommendation of the Prophet, peace and blessings be upon him.